A Combined Preconditioning Strategy for Nonsymmetric Systems

Abstract

Abstract. We present and analyze a class of nonsymmetric preconditioners within a normal (weighted least-squares) matrix form for use in GMRES to solve nonsymmetric matrix problems that typically arise in finite element discretizations. An example of the additive Schwarz method applied to nonsymmetric but definite matrices is presented for which the abstract assumptions are verified. A variable preconditioner, combining the original nonsymmetric one and a weighted least-squares version of it, is shown to be convergent and provides a viable strategy for using nonsymmetric preconditioners in practice. Numerical results are included to assess the theory and the performance of the proposed preconditioners.

Additive Schwarz with Variable Weights

“…In practical problems, the number of mesh points is very large, and thus also the number of unknowns in (1), and the resulting matrix is large and sparse. In these circumstances, iterative methods are often used, due to their ability to deal more effectively with a high degree of sparsity.…”

“…[1] 1 . What we propose in this paper is to go a step further, and implicitly find at each iteration both the current weights and all the weights at the previous iterations, so as to minimize the residual at the current step.…”

GMRES with multiple preconditioners

“…. , P −1 t r (0) ] ∈ R n×t , and consider the first iterate x (1) , which minimizes the residual norm of vectors over…”

“…Observe that Z (1) and V (2) have t columns, while Z (2) and V (3) have t 2 columns, and in general Z (i) and V (i+1) have t i columns. Therefore V k+1 has…”

“…In the recent paper [1], linear combinations of two preconditioners are used as in (2.12), and the coefficients are computed at each step to minimize the residual, as in MPGMRES. The difference is that only linear combinations (2.12) and not polynomials of higher order are sought.…”